222 Practical Mathematics

Differentiating again, and putting x = os we get

Similarly, we may find the values of the constants t, ff g . . . to as many terms
as we please by differentiating any number of times, and substituting x = o after
differentiation.

We find that the coefficients of all the even powers of x vanish, and that the
series is

i.2.3Ti.3.3-4-5 I-3. 3.4. 5. 6. 7 T ' ' '

We may use this series to calculate the value of sin x when x is known.
It is, of course, assumed in the differentiation of the above series that *
is measured in radians.

EXAMPLE (2).—-Show in the same -way that

COS X = I-----— 4-------------------------------------------- 4- .

1.2^1.2.3.4 1.2.3.4.5.6^'

EXAMPLE (3).—To calculate the -value of sin 10° correct to four decimal places.

We have 10° = 0^745 radian
.*. in the series

sin x = x —--------4. • x

•» « /> *

1.2.3 ' 3.2.3.4.5

we must take x = o* 1745.

It is most convenient to arrange the work as follows, so that each term can be
calculated from the preceding :-—

Positive Negative

terms. terms.

^ = 0-1745 =0*1745

X* = 0*03044
X* x X xy
_ _ —-j— = 0-1745 X 0-005073 » 0'000884

X*' ' x* X* x*

-——.- —------ —;-—-............- X — = —— X O-OOK22 =0"OOOOOI3

1.2.3.4.5 I . 2 . 3 ^ 20 1.2.3

X1 X* X* X*

................. ~e~ - = .............•......- X - - = —- X 0'00072 = O-OOOOOOIO

I.2.3.4.S-6.7 1.2.3.4.5*42 1.2.3.4.5 '

.'. sin 10° =0*1745 -0-00088
= 0-1736
to four decimal places.

We find from the tables that sin 10° = 0-1736. We see that the successive terms
in the series for sin x get smaller and smaller. We have omitted all terms beyond

_-----£_-_..__ assuming that their sum will not affect the fourth decimal place j

1.2. 3 . 4. 5 . 6 . f

this may be formally proved.

EXAMPLES.—LX VII.

1. Calculate the value of cos 10°, correct to four decimal places, and compare
with the tables.

2. Calculate tJae value of sin 5° correct to four decimal places, and compare witn

the tables.